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Test: Eigenvalues & Eigenvectors - 1


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20 Questions MCQ Test GATE Mechanical (ME) 2023 Mock Test Series | Test: Eigenvalues & Eigenvectors - 1

Test: Eigenvalues & Eigenvectors - 1 for Civil Engineering (CE) 2022 is part of GATE Mechanical (ME) 2023 Mock Test Series preparation. The Test: Eigenvalues & Eigenvectors - 1 questions and answers have been prepared according to the Civil Engineering (CE) exam syllabus.The Test: Eigenvalues & Eigenvectors - 1 MCQs are made for Civil Engineering (CE) 2022 Exam. Find important definitions, questions, notes, meanings, examples, exercises, MCQs and online tests for Test: Eigenvalues & Eigenvectors - 1 below.
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Test: Eigenvalues & Eigenvectors - 1 - Question 1

Find the sum of the Eigenvalues of the matrix

Detailed Solution for Test: Eigenvalues & Eigenvectors - 1 - Question 1

According to the property of the Eigenvalues, the sum of the Eigenvalues of a matrix is its trace that is the sum of the elements of the principal diagonal. 
Therefore, the sum of the Eigenvalues = 3 + 4 + 1 = 8.

Test: Eigenvalues & Eigenvectors - 1 - Question 2

Find the Eigenvalues of matrix

Test: Eigenvalues & Eigenvectors - 1 - Question 3

All the four entries of the 2 × 2 matrix    are nonzero, and one of its eigen values is zero. Which of the following statements is true?

Detailed Solution for Test: Eigenvalues & Eigenvectors - 1 - Question 3

One eigen value is zero

Test: Eigenvalues & Eigenvectors - 1 - Question 4

The eigen values of the following matrix are 

Detailed Solution for Test: Eigenvalues & Eigenvectors - 1 - Question 4

Let the matrix be A.  We know, Trace (A)=sum of eigen values. 

Test: Eigenvalues & Eigenvectors - 1 - Question 5

The three characteristic roots of the following matrix A  

Detailed Solution for Test: Eigenvalues & Eigenvectors - 1 - Question 5

A is lower triangular matrix. So eigen values are  only the diagonal elements. 

Test: Eigenvalues & Eigenvectors - 1 - Question 6

The sum of the eigenvalues of the matrix given below is

Detailed Solution for Test: Eigenvalues & Eigenvectors - 1 - Question 6

Sum of eigen values of  A= trace (A) 

Test: Eigenvalues & Eigenvectors - 1 - Question 7

For which value of x will the matrix given below become singular? 

Detailed Solution for Test: Eigenvalues & Eigenvectors - 1 - Question 7

Let the given matrix be A.  A is singular. 

Test: Eigenvalues & Eigenvectors - 1 - Question 8

Eigen values of a matrix    are 5 and 1. What are the eigen values of the matrix S2  = SS?

Detailed Solution for Test: Eigenvalues & Eigenvectors - 1 - Question 8

We know If λ be the eigen value of A ⇒λ2 is an eigen value of A2 .

Test: Eigenvalues & Eigenvectors - 1 - Question 9

The number of linearly independent eigenvectors of 

Detailed Solution for Test: Eigenvalues & Eigenvectors - 1 - Question 9

Number of linear independent vectors is equal to the sum of Geometric Multiplicity of eigen values. Here only eigen value is 2. To find Geometric multiplicity find n-r of (matrix-2I), where n is order and r is rank. Rank of obtained matrix is 1 and n=2 so n-r=1. Therefore the no of linearly independent eigen vectors is 1

Test: Eigenvalues & Eigenvectors - 1 - Question 10

The eigenvectors of the matrix     are written in the form  . What is a + b? 

Detailed Solution for Test: Eigenvalues & Eigenvectors - 1 - Question 10

 

Test: Eigenvalues & Eigenvectors - 1 - Question 11

One of the Eigenvectors of the matrix A =  is

Detailed Solution for Test: Eigenvalues & Eigenvectors - 1 - Question 11

The eigen vectors of A are given by  AX= λ X  

So we can check by multiplication.

  

Test: Eigenvalues & Eigenvectors - 1 - Question 12

The minimum and the maximum eigen values of the matrix    are –2 and 6, respectively. What  is the other eigen value?  

Detailed Solution for Test: Eigenvalues & Eigenvectors - 1 - Question 12

Test: Eigenvalues & Eigenvectors - 1 - Question 13

The state variable description of a linear autonomous system is, X= AX, 

Where X is the two dimensional state vector and A is the system matrix given by 

The roots of the characteristic equation are 

Detailed Solution for Test: Eigenvalues & Eigenvectors - 1 - Question 13

Characteristic equation will be :λ2 -4 =0 thus root of characteristic equation will be +2 and - 2.

Test: Eigenvalues & Eigenvectors - 1 - Question 14

For the matrix   s one of the eigen values is equal to -2. Which of the following  is an eigen vector? 

Detailed Solution for Test: Eigenvalues & Eigenvectors - 1 - Question 14

Test: Eigenvalues & Eigenvectors - 1 - Question 15

x=[x1x2…..xn]T is an n-tuple nonzero vector. The n×n matrix V=xxT    

Detailed Solution for Test: Eigenvalues & Eigenvectors - 1 - Question 15

As every minor of order 2 is zero. 

Test: Eigenvalues & Eigenvectors - 1 - Question 16

Cayley - Hamiltion Theorem states that square matrix satisfies its own characteristic equation, Consider a matrix 

A9 equals 

Detailed Solution for Test: Eigenvalues & Eigenvectors - 1 - Question 16

Test: Eigenvalues & Eigenvectors - 1 - Question 17

If the rank of a (5×6) matrix Q is 4, then which one of the following statements is correct?  

Detailed Solution for Test: Eigenvalues & Eigenvectors - 1 - Question 17

Rank of a matrix is equal to the No. of linearly independent row or no. of linearly  independent column vector. 

Test: Eigenvalues & Eigenvectors - 1 - Question 18

The trace and determinate of a 2 ×2 matrix are known to be – 2 and – 35 respectively. Its eigenvalues are 

Detailed Solution for Test: Eigenvalues & Eigenvectors - 1 - Question 18

Test: Eigenvalues & Eigenvectors - 1 - Question 19

Identify which one of the following is an eigenvector of the matrix  

Detailed Solution for Test: Eigenvalues & Eigenvectors - 1 - Question 19

Eigen Value (λ ) are 1,− 2.

 be the eigen  of A. Corresponding 

To λ then. 

be the eigen vector corrosponding to λ = 1

Test: Eigenvalues & Eigenvectors - 1 - Question 20

The eigenvalues of

Detailed Solution for Test: Eigenvalues & Eigenvectors - 1 - Question 20

The eigenvalues of an upper triangular matrix are simply the diagonal entries of the matrix.

Hence 5, -19, and 37 are the eigenvalues of the matrix. Alternately, look atd

λ = 5, -19, 37

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